Divide $$$x^{3} - 1$$$ by $$$1 - x^{2}$$$

Find $$$\frac{x^{3} - 1}{1 - x^{2}}$$$ using long division.

Write the problem in the special format (missed terms are written with zero coefficients):

$$$\begin{array}{r|r}\hline\\- x^{2}+1&x^{3}+0 x^{2}+0 x-1\end{array}$$$

Step 1

Divide the leading term of the dividend by the leading term of the divisor: $$$\frac{x^{3}}{- x^{2}} = - x$$$.

Write down the calculated result in the upper part of the table.

Multiply it by the divisor: $$$- x \left(- x^{2}+1\right) = x^{3}- x$$$.

Subtract the dividend from the obtained result: $$$\left(x^{3}-1\right) - \left(x^{3}- x\right) = x-1$$$.

Since the degree of the remainder is less than the degree of the divisor, we are done.

The resulting table is shown once more:

Therefore, $$$\frac{x^{3} - 1}{1 - x^{2}} = - x + \frac{x - 1}{1 - x^{2}}$$$.

$$$\frac{x^{3} - 1}{1 - x^{2}} = - x + \frac{x - 1}{1 - x^{2}}$$$A